Let Ω be a smooth bounded domain in ${\bf R}^n$, n > 1, let a and f be continuous functions on $\bar\Omega$, $1^\star = {n\over n-1}$. We are concerned here with the existence of solution in $BV(\Omega)$, positive or not, to the problem:


$$ \left\{ \begin{array}{rl} -{\rm div}\ \sigma+a(x) sign\ u &= f|u|^{1^\star-2} u\cr \sigma.\nabla u &= |\nabla u|\ {\rm in}\ \Omega\cr u\ {\rm is\ not \ identically\ zero}, &-\sigma.n (u) = |u|\ {\rm on }\ \partial \Omega.\end{array}\right.$$

This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$.